Remarks on a semilinear heat equation with integral boundary conditions

For a semilinear heat equation we consider a nonlocal boundary problem. On the basis of the solution of a Dirichlet problem for a parabolic equation and Volterra integral equation we establish the well-posedness for the nonlocal problem, which generalizes some recent results.     

On the solution of an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

Numerical solution of hyperbolic partial differential equation with an integral condition continues to be a major research area with widespread applications in modern physics and technology. Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary specifications have received much attention in last 20 years. However, most of the articles were directed to the second‐order parabolic equation, particularly to heat conduction equation. We will deal here with new type of nonlocal boundary value problem that is the solution of hyperbolic partial differential equations with nonlocal boundary specifications. These nonlocal conditions arise mainly when the data on the boundary can not be measured directly. Several finite difference methods have been proposed for the numerical solution of this one‐dimensional nonclassic boundary value problem. These computational techniques are compared using the largest error terms in the resulting modified equivalent partial differential equation. Numerical results supporting theoretical expectations are given. Restrictions on using higher order computational techniques for the studied problem are discussed. Suitable references on various physical applications and the theoretical aspects of solutions are introduced at the end of this article. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Global existence of solutions of initial boundary value problem for nonlocal parabolic equation with nonlocal boundary condition

We prove the global existence and blow-up of solutions of an initial boundary value problem for nonlinear nonlocal parabolic equation with nonlinear nonlocal boundary condition. Obtained results depend on the behavior of variable coefficients for large values of time.     

Composite spectral method for solution of the diffusion equation with specification of energy

Many physical subjects are modeled by nonclassical parabolic boundary value problems with nonlocal boundary conditions replacing the classic boundary conditions. In this article, we introduce a new numerical method for solving the one‐dimensional parabolic equation with nonlocal boundary conditions. The approximate proposed method is based upon the composite spectral functions. The properties of composite spectral functions consisting of terms of orthogonal functions are presented and are utilized to reduce the problem to some algebraic equations. The method is easy to implement and yields very accurate result. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Simultaneous Determination of Two Coefficients of a Parabolic Equation in the Case of Nonlocal and Integral Conditions

We establish conditions for the existence and uniqueness of a solution of the inverse problem for a parabolic equation with two unknown time-dependent coefficients in the case of nonlocal boundary conditions and integral overdetermination conditions.     

Blow-up for a degenerate parabolic equation with nonlocal source and nonlocal boundary

In this article, we investigate the blow-up properties of the positive solutions for a doubly degenerate parabolic equation with nonlocal source and nonlocal boundary condition. The conditions on the existence and nonexistence of global positive solutions are given. Moreover, we give the precise blow-up rate estimate and the uniform blow-up estimate for the blow-up solution.     

On One Nonlocal Problem with Free Boundary

We investigate group-theoretic properties of a nonlocal problem with free boundary for a degenerating quasilinear parabolic equation. We establish conditions for the invariant solvability of this problem, perform its reduction, and obtain an exact self-similar solution.     

The use of cubic B‐spline scaling functions for solving the one‐dimensional hyperbolic equation with a nonlocal conservation condition

Problems for parabolic partial differential equations with nonlocal boundary conditions have been studied in many articles, but boundary value problems for hyperbolic partial differential equations have so far remained nearly uninvestigated. In this article a numerical technique is presented for the solution of a nonclassical problem for the one‐dimensional wave equation. This method uses the cubic B‐spline scaling functions. Some numerical results are reported to support our study. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Necessary optimality conditions in an optimal control problem for a parabolic equation with nonlocal integral conditions

Doklady Mathematics - A variational method for the optimal control of moving sources governed by a parabolic equation with nonlocal integral conditions is considered. For this problem, an existence...     

Existence and approximation of nonlocal optimal design problems driven by parabolic equations

This work is a follow‐up to a series of articles by the authors where the same topic for the elliptic case is analyzed. In this article, a class of nonlocal optimal design problem driven by parabolic equations is examined. After a review of results concerning existence and uniqueness for the state equation, a detailed formulation of the nonlocal optimal design is given. The state equation is of nonlocal parabolic type, and the associated cost functional belongs to a broad class of nonlocal integrals. In the first part of the work, a general result on the existence of nonlocal optimal design is proved. The second part is devoted to analyzing the convergence of nonlocal optimal design problems toward the corresponding classical problem of optimal design. After a slight modification of the problem, either on the cost functional or by considering a new set of admissibility, the G‐convergence for the state equation and, consequently, the convergence of the nonlocal optimal design problem are proved.     

带非局部边界条件的非线性抛物方程的隐式差分解法

在准静态弹性力学中常遇到求解带有非局部边界条件的抛物方程初边值问题.本文构造了一个数值求解带有非局部边界条件的非线性抛物方程的隐式差分格式,利用离散泛函分析的知识和不动点定理证明了差分解是存在的,且在离散最大模意义下关于时间步长一阶收敛,关于空间步长二阶收敛,并给出了数值算例.     

Global existence of solutions for a p-Laplacian equation with nonlocal Fisher–KPP type reaction terms

This work is concerned with the Neumann initial boundary value problem and Cauchy problem of a parabolic p-Laplacian equation with nonlocal Fisher–KPP type reaction terms. We establish a uniform boundedness and global existence of solutions to the equation by applying the method of multipliers and modified Moser's iteration technique for some ranges of parameters. The ranges of parameters have similar structure to that of the classical critical Fujita exponent.     

Some Nonclassical Boundary Value Problems for Linear Mixed-Type Equations

Well-posedness of nonlocal boundary value problems is studied for some class of mixed-type equations which includes the Chaplygin equation and parabolic equations with varying time direction.     

Nonlocal boundary-value problems for abstract parabolic equations: well-posedness in Bochner spaces

The well-posedness of the nonlocal boundary-value problem for abstract parabolic differential equations in Bochner spaces is established. The first and second order of accuracy difference schemes for the approximate solutions of this problem are considered. The coercive inequalities for the solutions of these difference schemes are established. In applications, the almost coercive stability and coercive stability estimates for the solutions of difference schemes for the approximate solutions of the nonlocal boundary-value problem for parabolic equation are obtained.     

The eigenvalue problem for elliptic partial differential equation with multi-point nonlocal conditions

We study the spectral problem for the system of difference equations of a two-dimensional elliptic partial differential equation with nonlocal conditions. A new form of two-point nonlocal conditions that involve interior points is proposed. The matrix of the difference system is nonsymmetric thus different types of eigenvalues occur. The conditions for the existence of the eigenvalues and their corresponding eigenvectors are presented for the one dimensional problem. Then, these relations are generalized to the two-dimensional problem by the separation of variables technique.     

THE DECAY BEHAVIOR OF SOLUTIONS FOR NONLOCAL BOUNDARY PROBLEMS

A parabolic equation defined on a bounded domain with boundary condition being nonlocal is considered. The existence and the dynamic behavior of the solutions for linear and semilinear equations are investigated in special spaces. One will find that the behavior of the solutions are affected by the boundary conditions. A semigroup approach is employed in this paper.     

The operational matrices of Bernstein polynomials for solving the parabolic equation subject to specification of the mass

Some physical problems in science and engineering are modelled by the parabolic partial differential equations with nonlocal boundary specifications. In this paper, a numerical method which employs the Bernstein polynomials basis is implemented to give the approximate solution of a parabolic partial differential equation with boundary integral conditions. The properties of Bernstein polynomials, and the operational matrices for integration, differentiation and the product are introduced and are utilized to reduce the solution of the given parabolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new technique.     

An Explicit Solution of a Parabolic Equation with Nonlocal Boundary Conditions

We consider a parabolic differential equation with nonlocal boundary conditions. We find an explicit solution of the problem and prove the uniqueness of the solution. Analysis of the explicit solution reveals that it has three physically different linear components. The first component is of standing wave type, and the other two are of right- and left-going wave types, respectively. The speed of propagation of the heat waves depends on constants present in nonlocal boundary conditions. We give examples of right-going heat waves that have constant energy. This work was partially supported by Lithuanian State Science and Studies Foundation grant No. C-07/2003. Printed in Lietuvos Matematikos Rinkinys, Vol. 45, No. 3, pp. 315–332, July–September, 2005.     

Nonlocal parabolic problems in statistical mechanics

A nonlocal parabolic equation describing the evolution of a cloud of particles is studied.     

On an inverse problem of reconstructing a subdiffusion process from nonlocal data

We consider a problem of modeling the thermal diffusion process in a closed metal wire wrapped around a thin sheet of insulation material. The layer of insulation is assumed to be slightly permeable. Therefore, the temperature value from one side affects the diffusion process on the other side. For this reason, the standard heat equation is modified, and a third term with an involution is added. Modeling of this process leads to the consideration of an inverse problem for a one‐dimensional fractional evolution equation with involution and with periodic boundary conditions with respect to a space variable. This equation interpolates heat equation. Such equations are also called nonlocal subdiffusion equations or nonlocal heat equations. The inverse problem consists in the restoration (simultaneously with the solution) of the unknown right‐hand side of the equation, which depends only on the spatial variable. The conditions for overdefinition are initial and final states. Existence and uniqueness results for the given problem are obtained via the method of separation of variables.